Problem: $ 0.\overline{4} \div 0.\overline{37} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 4.4444...\\ x &= 0.4444...\end{align*} $ $\begin{align*} 9x &= 4 \\ x &= \dfrac{4}{9}\end{align*} $ $\begin{align*} 100y &= 37.3737...\\ y &= 0.3737...\end{align*} $ $\begin{align*} 99y &= 37 \\ y &= \dfrac{37}{99}\end{align*} $ So, the problem becomes: $ \dfrac{4}{9} \div \dfrac{37}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{4}{9} \times \dfrac{99}{37} = {?} $ $ \phantom{\dfrac{4}{9} \times \dfrac{37}{99}} = \dfrac{4 \times 99}{9 \times 37} $ $ \phantom{\dfrac{4}{9} \times \dfrac{37}{99}} = \dfrac{4 \times \cancel{99}11} {\cancel{9} \times 37} $ $ \phantom{\dfrac{4}{9} \times \dfrac{37}{99}} = \dfrac{44}{37} $